Week 8 - Game theory Flashcards
What does game theory use to study strategic situations?
mathematical models to study strategic situations
What are strategic situations based off?
your wellbeing not only depends on your behaviour, it depends on other peoples behaviour
What are two market structures use non-strategic situations?
- Perfect competition (firms price takers, dont have to worry about the prices of their competitors)
- Monopoly (no competitors to worry about)
What market structures use a strategic situation?
Everything in between a Monopoly and Perfect competition
What 2 interactions can Game theory be?
cooperative and competitive
When can an interaction be ‘cooperative’ in Game theory?
such as when business partners successfully collaborate on a project
When can an interaction be ‘competitive’ (or conflicting) in Game theory?
like when two or more firms fight for market share, rival politicians in an election, etc
What is Game theory a tool-kit to study?
conflict and cooperation between intelligent and rational decision makers
What are the two basic models of Games?
- Simultaneously played games (Normal form/ Strategic form)
- Sequentially played games (Extensive form)
What are Simultaneously played games (Normal form/ Strategic form)?
the games where the players have to choose their decisions or actions simultaneously
Simultaneous doesnt quite literally mean choosing at the same time, but means that players choose a decision without knowing what others are doing
What are sequentially played games (extensive form)?
moves are made sequentially (one after the other)
players can see clearly what moves have been taken so far before they make their own choice
What are three elements to a Normal form game?
- Players
- Strategies
- Payoffs/ Utilities
What is an example of a Normal form game matrix?
.
Player 2
M N
X a_1, a_2 b_1, b_2 Player 1
Y c_1, c_2 d_1, d_2
Players - have 2
Strategies - each player have 2
Outcomes - 4 in total (XM, XN, YM, YN)
a_1 represents player 1 payoff
a_2 represents player 2 payoff
What is a famous game in Game theory?
The Prisoner’s Dilemma
What is a version of the story The Prisoner’s Dilemma?
Two people (1&2) are arrested on charges of a bank robbery (major crime)
*Bad news: Evidence links them together as
collaborators and proves them to be at least
offenders (minor crime)
*Good news: they cannot prove that they are the real robbers, unless one of them confesses
*If one confesses and the other does not, the
confessor benefits from collaborating with the police and gets a lighter sentence
The police keep them in two separate cells
and give each of them the following choices
(Actions/Strategies):
1.Confess (“DEFECT” from the other prisoner)
2.Keep silent (“COOPERATE” with the other
prisoner)
*They are told that both are given the same
choices.
*Both will face the prison terms (Payoffs)
based on their actions
Eg1 The Prisoner’s Dilemma Matrix
.
Player 2
Cooperate Defect
Cooperate -1, -1 -4, 0
Player 1
Defect 0, -4 -3,-3
-1 = 1 month
can present with positive numbers
How do you solve a game?
Many ways, different economists have many slightly different ways of reaching the same result
Solutions must have the following qualities:
1. The outcome is stable (meaning nobody has an incentive to move from an outcome)
2. The outcome is reached by rational thinking (meaning players are assumed to think in a logical way to maximise their expected payoff
What is one of the main methods used to solve a game?
finding the best responses and using this to identify a Nash equilibrium
What are 3 approaches to solve a game and find a Nash equilibrium?
- Dominance
- Iterated Dominance
- ?
Strict dominance using The Prisoner’s Dilemma Matrix
.
Player 2
Cooperate Defect
Cooperate 4, 4 1, 5
Player 1
Defect 5, 1 2, 2
defect is always better than cooperate as the number is greater than cooperate
Defect, Defect is the Nash equilibrium, payoff 2,2
The outcome is Stable (meaning nobody has an incentive to move from this outcome)
What idea does dominance comes from?
that the reason we can delete actions are because other actions for the player exist that is always better
What is iterated dominance?
some games exist where actions can be ‘sequentially deleted’ because they are illogical to play if our agent is rational
iterated -> things that happen in ‘iterations’
(occur step by step
Eg of Iterated elimination of strictly dominated strategies
.
Player 2
L M R
U 13,3 1,4 7,3
Pla 1 M 4,1 3,3 6,2
D -1,9 2,8 8,-1
if rational Player 2 wont choose R
if rational Player 1 wont choose D, M better
if rational Player 2 M dominates L
Player 2 only chooses M
Player 1 wont choose U
then outcome MM 3,3
What does battle of the sexes have?
games having more than one Nash equilibrium
Examples of Battle of the sexes
You and your partner are planning a belated meal for Valentine’s Day; it’s been agreed that you will cook, whilst your partner will bring the wine.
You prefer beef and, naturally, red wine to go with it.
Your partner prefers fish and, naturally, white wine to go
with it. But you’ve both been really busy, and haven’t had a chance to talk about the meal beforehand
Your Partner
Red White
Beef 4,1 0,0 You
Fish 0,0 1,4
if my partner chooses red wine, pick beef (utility 4)
if my partner chooses white wine, pick fish (utility 1)
vise versa for the partner
so has 2 nash equilibrium
beef red 4,1 and fish white 1,4
both stable outcome
dont want utility of 0
What does Matching Pennies have?
Game with no (pure strategy) Nash equilibrium
Example of Matching pennies
Players 1 and 2 each put a penny on a table simultaneously. If the two pennies come up the same side (heads or tails) then player 1 gets both; otherwise player 2 does. We can represent this in the following matrix
Player 2
H T
H 1,-1 -1,1 Pla1
T -1,1 1,-1
if Player 2 chooses H, Player 1 will choose H
if Player 2 chooses T, Player 1 will choose T
if Player 1 chooses H, Player 2 will choose T
if Player 1 chooses T, Player 2 will choose H
in every outcome theres only one payoff highlighted, no nash equilibrium
also player can change strategy to other one, no stable outcome
doesnt have a pure strategy nash equilibrium
has a mixed strategy equilibrium
What sort of games are normal form games?
one shot games, the interaction happens just once
What sort of games are games likely to be in reality?
repeated games, the interaction happens many times or periodically
What are 2 types of repeated games?
- Finitely repeated games
- Infinitely repeated games
What are finitely repeated game?
the same (stage) game that is repeated a finite number of times
each prison dilemma is a stage game
in this repeated game (including eg 10 prison dilemmas)
a stage game looks like a table created before
What is an infinitely repeated game?
the same (stage) game that is repeated a infinite number of times
infinite amount of strategies
What are 2 strategies in infinitely repeated games?
- Grim trigger
- Tit for tat
What is the grim trigger strategy?
I will first cooperate. But as soon as you defect, I will defect for the remainder of the iterated game.
What is the tit for tat strategy?
I will first cooperate, then I will subsequently replicate your previous action. If you previously were cooperative, I am cooperative. If not, I am not.
Tit-for-tat has been very successfully used as a strategy for the repeated prisoner’s dilemma.
What are sequential games?
players who do not make decisions simultaneously, and one player’s decision affects the outcomes and decisions of other players
like chess
What are 2 types of sequential games?
- Perfect information games
- Imperfect information games
How do we model a sequential game (perfect information game)?
a game tree for an extensive form game (looks like upside down tree)
the normal form table doesnt tell you the structure of the story
Example of a perfect information game
eg Entry Deterrence
Two firms: Potential entrant (player 1) and Incumbent (player 2)
The entrant has two choices: Enter or Stay out (of the industry)
If entrant stays out, incumbent gets a large profit and entrant gets zero.
If the entrant decides to enter, then incumbent must choose whether to accommodate the entrant or enter
into a price war.
If incumbent starts a price war, then both firms suffer.
If incumbent accommodates the entrant, then they both obtain modest profits.
What is a backward induction?
starting with the last player’s action and moving backward. It effectively determines the Nash equilibrium for each subgame of the game
What is assumed in games?
people are selfish
What is behavioural game theory?
People are not always self-interested. For example, sometimes they may make decisions based on their social
preferences.
eg
Inequality aversion
Kindness reciprocity
Guilt aversion
People may not always best respond to each other. They may use other reasonings…
